A195679 Order of n-th homotopy group of the topological group O(oo), with -1 if the homotopy group is Z.

2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, -1

Offset: 0

Author

N. J. A. Sloane, Sep 22 2011

Keywords

• sign
• easy

Comments

Computed by R. Bott in 1957. Periodic with period 8.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• John C. Baez, The Octonions, Bull. Amer. Math. Soc., 39 (2002), 145-205.
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Crossrefs

Cf. A047530.

Programs

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{2, 2, 1, -1, 1, 1, 1, -1},128] (* Ray Chandler, Aug 25 2015 *)

PARI

Vec((1 + x + x^2)*(2 - x^2 + 2*x^4 - x^5) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)) + O(x^100)) \\ Colin Barker, Aug 28 2019

Formula

From Colin Barker, Aug 28 2019: (Start)

G.f.: (1 + x + x^2)*(2 - x^2 + 2*x^4 - x^5) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)).

a(n) = a(n-8) for n>7.

(End)

Extensions

Corrected by Harry Richman, Aug 27 2019